The question "What badges connect Hampshire and Lancashire?" likely references the historic naval association between Portsmouth (Hampshire) and Liverpool (Lancashire)—England's two greatest historic naval ports.
Both cities' coats of arms feature maritime heraldry:
County-level: Hampshire's rose and crown; Lancashire's red rose (Lancastrian). But the functional badge connecting the counties is the Royal Navy's White Ensign—flown in both ports for centuries.
If "badges" refers to military cap badges: Royal Hampshire Regiment (tiger) and King's/Lancashire Regiment (red rose) both amalgamated into Princess of Wales's Royal Regiment and Duke of Lancaster's Regiment respectively—now sharing the same brigade.